Gaussian Mixture Vector Autoregression∗

Gaussian Mixture Vector Autoregression Leena Kalliovirta Mika Meitz Pentti Saikkonen University of Helsinki University of Helsinki University of Helsinki Natural Resources Institute Finland January 5, 2016 Abstract This paper proposes a new nonlinear vector autoregressive (VAR) model referred to as the Gaussian mixture vector autoregressive (GMVAR) model. The GMVAR model belongs to the family of mixture vector autoregressive models and is designed for analyzing time series that exhibit regime-switching behavior. The main difference between the GMVAR model and previous mixture VAR models lies in the definition of the mixing weights that govern the regime probabilities. In the GMVAR model the mixing weights depend on past values of the series in a specific way that has very advantageous properties from both theoretical and practical point of view. A practical advantage is that there is a wide diversity of ways in which a researcher can associate different regimes with specific economically meaningful characteristics of the phenomenon modeled. A theoretical advantage is that stationarity and ergodicity of the underlying stochastic process are straightforward to establish and, contrary to most other nonlinear autoregressive models, explicit expressions of low order stationary marginal distributions are known. These theoretical properties are used to develop an asymptotic theory of maximum likelihood estimation for the GMVAR model whose practical usefulness is illustrated in a bi- variate setting by examining the relationship between the EUR—USDexchange rate and a related interest rate data. Keywords: mixture models, nonlinear vector autoregressive models, regime switching JEL Classification: C32 The authors thank the Academy of Finland (LK, MM, and PS), the OP-Pohjola Group Research Foundation (LK, MM, and PS), and Finnish Cultural Foundation (PS) for financial support. The paper has benefited from useful comments and suggestions made by the co-editor and three anonymous referees. Contact addresses: Leena Kalliovirta, Department of Political and Economic Studies, Univer- sity of Helsinki, P. O. Box 17, FI—00014 University of Helsinki, Finland, or Natural Resources Institute Finland (Luke), Viikinkaari 4, FI—00790 Helsinki, Finland; e-mail: leena.kalliovirta@luke.fi. Mika Meitz, Department of Political and Economic Studies, University of Helsinki, P. O. Box 17, FI—00014 Univer- sity of Helsinki, Finland; e-mail: mika.meitz@helsinki.fi. Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FI—00014 University of Helsinki, Finland; e-mail: pentti.saikkonen@helsinki.fi. 1 1 Introduction The vector autoregressive (VAR) model is one of the main tools used to analyze economic time series. Quite often, the VAR model is assumed linear, although both economic theory and previous empirical evidence may suggest that a nonlinear VAR model could be more appropriate. One popular nonlinear VAR model is the Markov switching VAR (MS—VAR) model that is designed to describe time series that switch between two or more regimes with each regime having the dynamics of a linear VAR model. In most applications, the regime switches are determined by a latent indicator variable that follows a time- homogeneous Markov chain with the transition probabilities depending on the most recent regime but not on past observations (see, e.g., Krolzig (1997) and Sims, Waggoner, and Zha (2008)). More general time-inhomogeneous MS—VAR models, where the transition probabilities depend both on the most recent regime and on past observations, have also been considered (see, e.g., Ang and Bekaert (2002)). In this paper, we are interested in mixture VAR (MVAR) models. These models can be viewed as special cases of general time-inhomogeneous MS—VAR models from which they are obtained with suitable parameter restrictions. They differ from the commonly used time-homogeneous MS—VAR models in that the transition probabilities do not depend on the most recent regime, but instead on past observations. An equivalent formulation of MVAR models (explaining the nomenclature ‘mixture’) is to specify the conditional distribution of the process as a mixture of (typically) Gaussian conditional distributions of linear VAR models. Different models are obtained by different specifications of the mixing weights. Univariate mixture autoregressive models were introduced by Le, Martin, and Raftery (1996) and further developed by Wong and Li (2000, 2001a,b) (for further references, see Kalliovirta, Meitz, and Saikkonen (2015); for mixture autoregressions in Bayesian framework, see, e.g., Villani, Kohn, and Giordani (2009)). Extensions to the vector case with economic applications involving inflation, interest rates, stock prices, and exchange rates have been presented by Lanne (2006), Fong, Li, Yau, and Wong (2007), Bec, Rahbek, and Shephard (2008), and Dueker, Psaradakis, Sola, and Spagnolo (2011). In this paper, we propose a new mixture VAR model referred to as the Gaussian mixture vector autoregressive (GMVAR) model. This model is a multivariate generalization of a similar univariate model introduced in Kalliovirta et al. (2015). The specific formulation of the GMVAR model turns out to have very convenient theoretical implications. To highlight this point, first recall a property that makes the stationary linear Gaussian VAR model different from most, if not nearly all, of its nonlinear alternatives, namely that 2 the probability structure of the underlying stochastic process is fully known and can be described by Gaussian densities. In nonlinear VAR (and also nonlinear AR) models the situation is typically very different: the conditional distribution is known by construction but what is usually known beyond that is only the existence of a stationary distribution and finiteness of some of its moments. In the GMVAR model, stationarity of the underlying stochastic process is a simple consequence of the definition of the model. Moreover, letting p denote the autoregressive order of the model (see Section 2), the stationary distribution of p + 1 consecutive (vector valued) observations is known to be a mixture of multivariate Gaussian distributions with constant mixing weights and known structure for the mean and covariance matrix of the component distributions, whereas the conditional distribution is a multivariate Gaussian mixture with time varying mixing weights. Thus, similarly to the linear Gaussian VAR model, and contrary to (at least most) other nonlinear VAR models, the structure of stationary marginal distributions consisting of p + 1 observations or less is fully known in the GMVAR model. In order to interpret a multivariate regime-switching model one typically aims at asso- ciating different economically meaningful regimes with different states of economic vari- ables, such as high or low level of inflation, interest rate, or asset return. An appealing feature of the GMVAR model is that, due to the specific structure of the mixing weights, the researcher can associate different regimes with different characteristics of the phenomenon modeled. Moreover, in the GMVAR model switches between regimes are allowed to depend not only on, say, the level of past observations, but on their entire distribution. Thus, in addition to regime switches taking place in periods of high/low levels of the considered series, the GMVAR model can also allow for regime switches taking place in periods of high/low variability, or high/low temporal dependence, and combinations of all these. These convenient features are illustrated in our empirical example, which also demonstrates promising forecasting power of the GMVAR model. We believe that introducing the GMVAR model makes a useful addition to the lit- erature on multivariate regime-switching models. This is mainly due to the formulation of the model which, in addition to the attractive properties already discussed, has the following implications. First, the regime-switching mechanism is parsimonious and its form becomes automatically specified once the number of regimes and the order of the model are chosen; there is no need to find out which lagged values of the considered series are used to model the regime-switching mechanism and in what form they should be included in the model. Second, conditions that guarantee stationarity (and ergodicity) of the model are entirely similar to those in linear VAR models and they are also necessary 3 in the sense of not overly restricting the parameter space of the model. These conditions are therefore both sharp and easy to check, and there is no need to use simulation to find out whether an estimated model fulfills the stationarity condition. The plan of the paper is as follows. Section 2 discusses general mixture VAR models. Section 3 introduces the new GMVAR model, discusses its theoretical properties, and establishes the consistency and asymptotic normality of the maximum likelihood es- timator. Section 4 presents an empirical example with exchange rate and interest rate data, discusses issues of model building, and compares the forecasting performance of the GMVAR model to other linear and nonlinear VAR models. Section 5 concludes, and an Appendix contains some technical derivations. A ‘Supplementary Appendix’(available from the authors) contains additional material omitted from the paper. Finally, a word on notation. We use vec (A) to denote a column vector obtained by stacking the columns of the matrix A one below another. If A is a symmetric matrix then vech (A) is a column vector obtained by stacking the columns of A from the principal diagonal downwards (including elements on the diagonal). The usual notation A B is used for the Kronecker product of the matrices A and B. To simplify notation, we shall write z = (z1, . , zm) for the (column) vector z where the components zi may be either scalars or vectors (or both). For any scalar, vector, or matrix x, the Euclidean norm is denoted by x . j j 2 Multivariate mixture autoregressive models Let yt (t = 1, 2,...) be the d—dimensional time series of interest, and let t 1 = (ys, s < t) F denote the —algebra generated by past yt’s. We use Pt 1 ( ) to signify the conditional · probability of the indicated event given t 1.

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